(a) Using the Bohr's model, calculate the speed of the electron in a hydrogen atom in the n=1,2 and 3 levels. (b) Calculate the orbital period in each of these levels.

Explanation

(a) Speed of the electron in the Bohr's nth orbit`v=c/va` where, `a =(2piKe^(2))/(ch)`
` a=0.0073 thereforev=c/nxx0.0073`
for `n=1, v_(1)=c/1xx0.0073`
`=3xx10^(8)xx0.0073=2.19xx10^(6)ms`
for `n=2 v_(2)=c/2xx0.0073`
=`(3xx10^(8)xx0.0073)/2=1.095xx10^(6)m//s`
For `n=3 v_(3) c/3xx0.0073`
` =(3xx10^(8)xx0.0073)/3 =7.3xx10^(5)m//s`
orbit period of electron is given by` T=(2pir)/v`
Radius of nth orbit `r_(n)=(n^(2)/m)(h/(2pi))^(2) (4piepsilon_(0))/e^(2)`
` thereforer_(1)=((1)^(2)xx(6.63xx10^(-34)))/(4xx9.87xx(9xx10^(9))xx9xx10^(-31)xx6.6xx10^(-19))`
` = 0.53xx10^(10)m`.
` n=1 T_(1)=(2pir_(1))/v_(1)`
`(2xx3.14xx0.53xx10^(-10))/(2.19xx10^(6)= 1.52xx10^(-16)s`
For `n=2`, radius `r_(n)=n^(2)r_(1)`
`r_(2)=2^(2). r=4xx0.53xx10^(-10) m` and
Velocity `v_(n)=v_(1)/n`
` therefore v_(2)=v^(1)/2=(2.19xx10)/2` time period `T_(2) =(2xx3.14xx4xx0.53xx10^(-10)xx2)/(2.19xx10^(6))`
`= 1.216xx10^(-15)s.`
for `n=3`, radius `r_(3) =3^(2)`
` r_(1)=9r_(1)=9xx0.53xx10^(-10)m`
and velocity `v_(3)=vv^(1)/3=(2.19xx10^(6)/3m//s`
Time Period `T_(3)=(2pir_(3))/v_(3)`
`= (2xx3.14xx9xx0.53xx10_(-10)xx^(-10)xx3)/2.19xx10^(6)=4.1xx10^(-15)s`.